Transcript mean - Shelton State Community College

Chapter 9
Statistics
Section 9.1
Frequency Distributions;
Measures of Central Tendency
Random Samples
When a characteristic of a population
needs to be studied, it is sometimes not
possible to examine all the elements in the
population.
 A limited sample is used when the
population is too large.
 In order for the inferences gained from
the study to be correct, the sample chosen
must be a random sample.

Random Samples
Random samples are representative of the
population because they are chosen so
that every element of the population is
equally likely to be selected.
 Often difficult to obtain in real life.
 Once a sample has been chosen and all
data collected, the data must be organized
so that conclusions may be more easily
drawn.

Organizing Data
One method of organizing data is to group
the data into intervals (usually equal
intervals).
 A grouped frequency distribution is a table
that displays each interval and the number
of times data points occur in the intervals.

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College Units in Management a
Business Major Should Have
Graphically Organizing Data
The information in a grouped frequency
distribution can be displayed in a
histogram similar to the histograms for
probability distributions.
 The intervals determine the widths of the
bars. (Equal intervals = equal bar widths)
 The heights of the bars are determined by
the frequencies.

Frequency Polygons
A frequency polygon is another form of
graph that illustrates a grouped frequency
distribution.
 The polygon is formed by joining
consecutive midpoints of the tops of the
histogram bars with straight line
segments.
 The midpoints of the first and last bars are
joined to endpoints on the horizontal axis
where the next midpoint would appear.

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College Units in Management a Business
Major Should Have
Example 1

Trooper Barney Fife recorded the following
speeds along Mayberry Highway during a 1 hour
period. All speeds are mph.
45
39
50
39
46
47
55
61
29
42
54
57
52
33
60
Construct a frequency distribution, histogram, and
frequency polygon for this information.
Measures of Central Tendency
Three measures of central tendency, or
“averages” are used with frequency
distributions to describe the data.
Mean
Median
Mode
Mean
Most important, and commonly used, measure of
central tendency.
 The arithmetic mean (the mean) of a set of numbers
is the sum of the numbers, divided by the total
number of numbers.

Example 2
Help Trooper Fife find the average speed
along Mayberry Highway during that
particular hour.
45
39
50
39
46
47
55
61
29
42
54
57
52
33
60
Mean for Frequency Distributions
Example 3
Use the frequency distribution for Trooper
Fife’s data to determine the mean.
 Compare this mean to the mean of the
ungrouped data.

Two Types of Mean

Sample mean: mean of a random sample. This
mean is used most often when the population is
very large.

Population mean: mean for the entire
population. The expected value of a random
variable in a probability distribution is sometimes
called the population mean. Denoted by µ.
(Greek letter “mu”)
Median
The median is the middle entry in a set of
data arranged in either increasing or
decreasing order.
 If there is an even number of entries, the
median is defined to be the mean of the
two center entries.

Statistic
Both the mean and median are examples
of a statistic, which is simply a number
that gives information about a sample.
 Sometimes the median gives a truer
representation or typical element of the
data than the mean.
 The mean is sometimes not the best
representation because it is easily
influenced by extreme outliers.

Mode
The mode is the most frequently occurring
entry, or entries, in a set of data.
 If there is no one entry that occurs more
than the others we say there is no mode.
 Sometimes, the data set will have more
than one mode.

Example 4
Find the speed along Mayberry Highway
that would represent the median speed.
What speed(s) represent the mode, if any?
45
39
50
39
46
47
55
61
29
42
54
57
52
33
60
Advantages and Disadvantages for Mean,
Median, and Mode
Measure of
Central
Tendency
Advantage
Disadvantage
Easy
Mean
to compute
Influenced by extreme
Takes all data values into account
values
Reliable
Easy
Median
to compute
Not influenced by extreme values Difficult to rank large
number of data values
Can be used with non-numerical
data
Easily
Mode
found
Not influenced by extreme values Can’t always locate
just one mode
Can be used with non-numerical
data